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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Injectivity of quasi-isometric mappings of balls
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by Julian Gevirtz PDF
Proc. Amer. Math. Soc. 85 (1982), 345-349 Request permission

Abstract:

Let $X$ and $Y$ be real Banach spaces. A mapping $f$ of an open subset $R$ of $X$ into $Y$ is said to be $(m,M)$-isometric if it is a local homeomorphism for which $M \geqslant {D^ + }f(x)$ and $m \leqslant {D^ - }f(x)$ for all $x$ in $R$, where ${D^ + }f(x)$ and ${D^ - }f(x)$ are, respectively, the upper and lower limits of $|f(y) - f(x)|/|y - x|$ as $y \to x$. We show that if $R$ is a ball then all $(m,M)$isometric mappings of $R$ are injective provided that $M/m < 1.114 \ldots$ and we also give some numerical improvements of similar results of F. John for the special case that $X$ is a Hilbert space.
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Additional Information
  • © Copyright 1982 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 85 (1982), 345-349
  • MSC: Primary 47H99; Secondary 46B99
  • DOI: https://doi.org/10.1090/S0002-9939-1982-0656099-X
  • MathSciNet review: 656099